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<p>The <dfn class="terminology">radius of convergence</dfn> (about <span class="process-math">\(x_0\)</span>): a nonnegative <span class="process-math">\(\textcolor{blue}{\rho}\text{,}\)</span> such that</p>
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\begin{equation*}
\sum_{n=0}^{\infty} a_n(x-x_0)^n ~~\left\{\begin{array}{ll} \text{\textcolor{blue}{converges absolutely}} &amp; \text{for $\textcolor{magenta}{|x-x_0|&lt;}\textcolor{blue}{\rho}$},\\ \text{diverges} &amp;\text{for $|x-x_0|&gt;\textcolor{blue}{\rho}$.}\end{array}\right.
\end{equation*}
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